\subsection{more thoughts on variational equations}
Let us assume at the BEC side, $\mu$ is negative and large, so $\xi_k$ does not change much, this leads to a constant $\theta_k,\phi_k$ from eqs. (\ref{eq:tan1},\ref{eq:tan2}).  This does not sounds correctly as there probably should be some varies over $k$ for $\phi_k$, i.e., the weight between open and close channels.  There are several places unclear.  1.  $\Delta_k$ might not be constant here over $\epsilon_F$;  2. how to compare Fermi energy $\epsilon_F$, $\mu$ and $\eta$?   Well, answer to the second questions seems to be $|\mu|\sim{}E_{binding}\ll|\eta|$ for cloes to resonance (within $\delta_c$) and $|\mu|\sim{}E_{binding}\sim|\eta|$ far away.  Also $|\eta|\ll\epsilon_F$ for narrow resonance.   


Another problem, $\eta$ in \ref{sec:variation} seems to be the detuning from the absolute value instead of the resonance.  This does not matter much for the broad resonance, as that is corresponding large $\widetilde\eta$; but seems be more problemetic when interpretating it in the narrow resonance.  